A well-known theorem states that if $f(z)$ generates a PF$_r$ sequence then $1/f(-z)$ generates a PF$_r$ sequence. We give two counterexamples which show that this is not true, and give a correct version of the theorem. In the infinite limit the result is sound: if $f(z)$ generates a PF sequence then $1/f(-z)$ generates a PF sequence.

Keywords:

total positivity, Toeplitz matrix, Pólya frequency sequence, skew Schur function total positivity, Toeplitz matrix, Pólya frequency sequence, skew Schur function

MSC Classifications:

15A48, 15A45, 15A57, 05E05 show english descriptions Positive matrices and their generalizations; cones of matrices
Miscellaneous inequalities involving matrices
Other types of matrices (Hermitian, skew-Hermitian, etc.)
Symmetric functions and generalizations 15A48 - Positive matrices and their generalizations; cones of matrices
15A45 - Miscellaneous inequalities involving matrices
15A57 - Other types of matrices (Hermitian, skew-Hermitian, etc.)
05E05 - Symmetric functions and generalizations

top of page | contact us | social media | privacy | site map